Theorem 2.

Proof.

As described above, if SSS is a set of prime factors
of an abundant number, then we may bound each term
in the inequality of the previous theorem to obtain
the inequality in the current theorem. Assume, then
that SSS is a finite set of prime numbers which
satisfies said inequality. Then, by continuity,
there must exist a real numberϵ>0ϵ0\epsilon>0 such
that